Alright, let's solve this step by step.
Given:
1. The age difference between two brothers is 4 years.
2. The product of their ages is 221.
Let's denote:
- [tex]\( x \)[/tex] as the age of the younger brother.
- [tex]\( x + 4 \)[/tex] as the age of the older brother, since he is 4 years older.
From the given information, we can set up the equation:
[tex]\[ x \times (x + 4) = 221 \][/tex]
Now, we expand and rearrange this equation:
[tex]\[ x^2 + 4x = 221 \][/tex]
To solve this quadratic equation, we rearrange it into standard form:
[tex]\[ x^2 + 4x - 221 = 0 \][/tex]
To find the roots of this quadratic equation, we use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
In our equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -221 \)[/tex]. Plugging these values into the quadratic formula:
[tex]\[ x = \frac{{-4 \pm \sqrt{{4^2 - 4 \cdot 1 \cdot (-221)}}}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{{-4 \pm \sqrt{{16 + 884}}}}{2} \][/tex]
[tex]\[ x = \frac{{-4 \pm \sqrt{900}}}{2} \][/tex]
[tex]\[ x = \frac{{-4 \pm 30}}{2} \][/tex]
This gives us two possible solutions for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{{-4 + 30}}{2} = \frac{26}{2} = 13 \][/tex]
[tex]\[ x = \frac{{-4 - 30}}{2} = \frac{-34}{2} = -17 \][/tex]
Since age cannot be negative, we discard [tex]\( x = -17 \)[/tex]. Hence, we have:
- The age of the younger brother is [tex]\( x = 13 \)[/tex].
To find the age of the older brother, we add 4 years to the younger brother's age:
[tex]\[ 13 + 4 = 17 \][/tex]
So, the ages of the brothers are:
- The younger brother is 13 years old.
- The older brother is 17 years old.
